September  2011, 1(3): 307-330. doi: 10.3934/mcrf.2011.1.307

Global Carleman estimate on a network for the wave equation and application to an inverse problem

1. 

CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4, France

2. 

Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin en Yvelines, F-78035 Versailles, France

3. 

Institut Elie Cartan de Nancy & INRIA (Project-Team CORIDA), BP 239, F-54506 Vandœuvre-les-Nancy Cedex, France

Received  March 2011 Revised  May 2011 Published  September 2011

We are interested in an inverse problem for the wave equation with potential on a star-shaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.
Citation: Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395.

[2]

S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings,, ZAMM Z. Angew. Math. Mech., 90 (2010), 136. doi: 10.1002/zamm.200900295.

[3]

L. Baudouin, A. Mercado and A. Osses, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem,, Inverse Problems, 23 (2007), 257. doi: 10.1088/0266-5611/23/1/014.

[4]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647. doi: 10.1088/0266-5611/20/3/002.

[5]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients,, Appl. Anal., 83 (2004), 983. doi: 10.1080/0003681042000221678.

[6]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem,, J. Math. Anal. Appl., 336 (2007), 865. doi: 10.1016/j.jmaa.2007.03.024.

[7]

A. L. Bukhgeim, "Volterra Equations and Inverse Problems,", Inverse and Ill-Posed Problems Series, (1999).

[8]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Dokl. Akad. Nauk SSSR, 260 (1981), 269.

[9]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590. doi: 10.1137/S0363012903421844.

[10]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, In, (2000), 1006.

[11]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures," Mathématiques & Applications (Berlin), 50,, Springer-Verlag, (2006).

[12]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157. doi: 10.1088/0266-5611/19/1/309.

[13]

Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," second edition, Applied Mathematical Sciences, 127,, Springer, (2006).

[15]

M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[16]

M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data,, Inverse Problems, 7 (1991), 577. doi: 10.1088/0266-5611/7/4/007.

[17]

M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse and Ill-Posed Problems Series, (2004).

[18]

J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures,", Systems & Control: Foundations & Applications, (1994).

[19]

I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. doi: 10.1006/jmaa.1999.6348.

[20]

J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte," Recherches en Mathématiques Appliquées, 8,, Masson, (1988).

[21]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Translated from the French by P. Kenneth, (1972).

[22]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425.

[23]

S. Nicaise and O. Zaïr, Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees,, Rev. Mat. Complut., 16 (2003), 151.

[24]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55. doi: 10.1515/jiip.1997.5.1.55.

[25]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, SIAM J. Control Optim., 30 (1992), 229. doi: 10.1137/0330015.

[26]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590.

[27]

F. Visco-Comandini, M. Mirrahimi and M. Sorine, Some inverse scattering problems on star-shaped graphs,, J. Math. Anal. Appl., 378 (2011), 343.

[28]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl. (9), 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.

[29]

M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns,, Appl. Math. Optim., 48 (2003), 211. doi: 10.1007/s00245-003-0775-5.

[30]

X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812. doi: 10.1137/S0363012999350298.

[31]

E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011).

show all references

References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395.

[2]

S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings,, ZAMM Z. Angew. Math. Mech., 90 (2010), 136. doi: 10.1002/zamm.200900295.

[3]

L. Baudouin, A. Mercado and A. Osses, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem,, Inverse Problems, 23 (2007), 257. doi: 10.1088/0266-5611/23/1/014.

[4]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647. doi: 10.1088/0266-5611/20/3/002.

[5]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients,, Appl. Anal., 83 (2004), 983. doi: 10.1080/0003681042000221678.

[6]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem,, J. Math. Anal. Appl., 336 (2007), 865. doi: 10.1016/j.jmaa.2007.03.024.

[7]

A. L. Bukhgeim, "Volterra Equations and Inverse Problems,", Inverse and Ill-Posed Problems Series, (1999).

[8]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Dokl. Akad. Nauk SSSR, 260 (1981), 269.

[9]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590. doi: 10.1137/S0363012903421844.

[10]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, In, (2000), 1006.

[11]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures," Mathématiques & Applications (Berlin), 50,, Springer-Verlag, (2006).

[12]

O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157. doi: 10.1088/0266-5611/19/1/309.

[13]

Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," second edition, Applied Mathematical Sciences, 127,, Springer, (2006).

[15]

M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[16]

M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data,, Inverse Problems, 7 (1991), 577. doi: 10.1088/0266-5611/7/4/007.

[17]

M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse and Ill-Posed Problems Series, (2004).

[18]

J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures,", Systems & Control: Foundations & Applications, (1994).

[19]

I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. doi: 10.1006/jmaa.1999.6348.

[20]

J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte," Recherches en Mathématiques Appliquées, 8,, Masson, (1988).

[21]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Translated from the French by P. Kenneth, (1972).

[22]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425.

[23]

S. Nicaise and O. Zaïr, Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees,, Rev. Mat. Complut., 16 (2003), 151.

[24]

J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55. doi: 10.1515/jiip.1997.5.1.55.

[25]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, SIAM J. Control Optim., 30 (1992), 229. doi: 10.1137/0330015.

[26]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590.

[27]

F. Visco-Comandini, M. Mirrahimi and M. Sorine, Some inverse scattering problems on star-shaped graphs,, J. Math. Anal. Appl., 378 (2011), 343.

[28]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl. (9), 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.

[29]

M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns,, Appl. Math. Optim., 48 (2003), 211. doi: 10.1007/s00245-003-0775-5.

[30]

X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812. doi: 10.1137/S0363012999350298.

[31]

E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011).

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